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\section*{commutative algebraic theory}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include higher algebra - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{abstract_formulation}{Abstract formulation}\dotfill \pageref*{abstract_formulation} \linebreak
\noindent\hyperlink{commutative_theory_as_monoidal_monad}{Commutative theory as monoidal monad}\dotfill \pageref*{commutative_theory_as_monoidal_monad} \linebreak
\noindent\hyperlink{as_commutative_monoid_in_a_duoidal_category}{As commutative monoid in a duoidal category}\dotfill \pageref*{as_commutative_monoid_in_a_duoidal_category} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{ClosedMonoidalStructureOnAlgebras}{Closed monoidal structure on algebras}\dotfill \pageref*{ClosedMonoidalStructureOnAlgebras} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

An \emph{[[algebraic theory]]} is said to be \emph{commutative} if its operations are algebra homomorphisms under any interpretation, generalizing the familiar case of the theory of [[commutative monoid|commutative monoids]].

A more general notion is that of \emph{[[monoidal monads]]}.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Two operations, $\alpha$ and $\beta$, of an [[algebraic theory]] are said to \textbf{commute} if for any [[matrix]] $M$ of elements, with the number of rows given by the arity of $\alpha$ and the number of columns by the arity of $\beta$, one gets the same result whether one

\begin{enumerate}%
\item applies $\alpha$ to each column of $M$ and then $\beta$ to the resulting row, or
\item applies $\beta$ to each row of $M$ and then $\alpha$ to the resulting column.

\end{enumerate}
(We formulate this notion in an element-free way below.)

Note that an operation of arity $0$ or $1$ always commutes with itself; this is not necessarily the case for higher arities. Commuting nullary operations are necessarily equal.

The operations that commute with a given set of operations in an algebraic theory form a subtheory. The \textbf{centre} of an algebraic theory is given by the operations that commute with all the operations of the theory. An algebraic theory is \textbf{commutative} if every pair of its operations commute. Another way of describing the centre is to say that it consists of those operations which are also [[homomorphisms]]; an algebraic theory is commutative if all of its operations are homomorphisms.

Here is a more formal definition, expressed in terms of structure on the monad $T \colon Set \to Set$ associated with the algebraic theory.

\begin{def}
\label{}\hypertarget{}{}
$T$ is a \emph{[[commutative monad]]} if there is an equality between two maps

\begin{displaymath}
\alpha = \beta \colon T A \times T B \stackrel{\to}{\to} T(A \times B)
\end{displaymath}
where

\begin{itemize}%
\item $\alpha$ is the composite

\begin{displaymath}
T A \times T B \stackrel{\sigma_{A, T B}}{\to} T(A \times T B) \stackrel{T(\tau_{A, B})}{\to} T T(A \times B) \stackrel{m(A \times B)}{\to} T(A \times B).
\end{displaymath}
Here $\sigma$ denotes the \emph{strength} of the monad $T$, and $\tau$ its symmetric counterpart.


\item $\beta$ is the composite

\begin{displaymath}
T A \times T B \stackrel{\tau_{T A, B}}{\to} T(T A \times B) \stackrel{T(\sigma_{A, B})}{\to} T T(A \times B) \stackrel{m(A \times B)}{\to} T(A \times B).
\end{displaymath}


\end{itemize}
\end{def}
It is worth checking what this description gives more explicitly. Starting with a pair of elements in $T A \times T B$,

\begin{displaymath}
\langle (\omega; a_1, \ldots, a_m), (\chi; b_1, \ldots, b_n)\rangle
\end{displaymath}
(and here we should be considering equivalence classes of such formal operations), the map $\alpha$ sends this to

\begin{displaymath}
(\omega(\chi, \ldots, \chi); (a_1, b_1), \ldots (a_1, b_n)), \ldots, (a_m, b_1), \ldots, (a_m, b_n))
\end{displaymath}
where $\omega(\chi, \ldots, \chi)$ is the evident operation of arity $m n = n + \ldots + n$. In more detail, $\alpha(\langle (\omega; \vec{a}), (\chi; \vec{b} \rangle)$ is the end result of the sequence

\begin{displaymath}
\langle (\omega; \vec{a}), (\chi; \vec{b}) \rangle \stackrel{\sigma}{\mapsto} (\omega; (a_1, (\chi; \vec{b})), \ldots, (a_m, (\chi; \vec{b}))) \stackrel{T\tau}{\mapsto} (\omega; (\chi; (a_1, \vec{b})), \ldots, (\chi; (a_m, \vec{b}))) \stackrel{m}{\mapsto} (\omega(\chi, \ldots, \chi); (a_1, \vec{b}), \ldots, (a_m, \vec{b})).
\end{displaymath}
Similarly, the map $\beta$ sends the pair $\langle (\omega; \vec{a}), (\chi; \vec{b}) \rangle$ to

\begin{displaymath}
(\chi(\omega, \ldots, \omega); (\vec{a}, b_1), \ldots, (\vec{a}, b_n))
\end{displaymath}
where $\chi(\omega, \ldots, \omega)$ is the evident operation of arity $n m = m + \ldots + m$.

\hypertarget{abstract_formulation}{}\subsubsection*{{Abstract formulation}}\label{abstract_formulation}

The category $\mathbf{Th}$ of Lawvere theories is endowed with a symmetric monoidal product (called \emph{[[tensor product theory|Kronecker product]]}; see Freyd's article in the references),

\begin{displaymath}
\otimes \colon \mathbf{Th} \times \mathbf{Th} \to \mathbf{Th},
\end{displaymath}
whereby $(S \otimes T)$-algebras are $S$-algebras internal to $T$-algebras, or equally well $T$-algebras internal to $S$-algebras. A commutative theory is tantamount to a commutative monoid in the symmetric monoidal category $\mathbf{Th}$.

If $S$ and $T$ are commutative theories, then their coproduct in the category of commutative theories is $S \otimes T$.

\hypertarget{commutative_theory_as_monoidal_monad}{}\subsubsection*{{Commutative theory as monoidal monad}}\label{commutative_theory_as_monoidal_monad}

Let $T$ be the $Set$-monad of a commutative theory. Then the map

\begin{displaymath}
\alpha_{A, B}: T A \times T B \to T(A \times B)
\end{displaymath}
as defined above can be shown to be the structure map for a monoidal structure on $T$, i.e., making $T$ a lax (symmetric) monoidal functor, and in fact the monad multiplication and unit become monoidal transformations. In other words, we get a monad in the 2-category of symmetric monoidal categories, lax symmetric monoidal functors, and monoidal transformations: a [[monoidal monad]].

In fact, it may be shown that commutative Lawvere theories on $Set$ are precisely the same things as (finitary) symmetric monoidal monad structures on $(Set, \times)$, as shown by Anders Kock. For more on this, see [[monoidal monad]].

\hypertarget{as_commutative_monoid_in_a_duoidal_category}{}\subsubsection*{{As commutative monoid in a duoidal category}}\label{as_commutative_monoid_in_a_duoidal_category}

The commutativity of a theory can also be expressed as an abstract property of a monoid in a [[duoidal category]], specialized to the duoidal category of finitary endofunctors.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{ClosedMonoidalStructureOnAlgebras}{}\subsubsection*{{Closed monoidal structure on algebras}}\label{ClosedMonoidalStructureOnAlgebras}

We discuss that the category of [[algebras for an algebraic theory]] over a commutative algebraic theory is canonically a [[closed monoidal category|closed]] [[symmetric monoidal category]] (\hyperlink{Keigher78}{Keigher 78}, \hyperlink{Seal12}{Seal 12}).

If $f_1,\ldots , f_n$ are homomorphisms $A \to B$ of models (algebras) of a commutative algebraic theory, and $\omega$ is an $n$-ary operation of it, then the function $A \to B$ given by sending $a \in A$ to $\omega(f_1(a),\ldots ,f_n(a)) \in B$ is again a homomorphism, which is naturally called $\omega(f_1,\ldots ,f_n)$. In this way $Hom(A,B)$ is enriched as a model of the algebraic theory, and we have a [[closed category]] of models and homorphisms. Furthermore, this [[internal hom|internal]] $Hom$ has a [[left adjoint]] $\otimes$ for which the free model on one generator is a unit, so we have a [[closed monoidal category]], in fact a closed [[symmetric monoidal category]].

The monoidal structure $\otimes$ can be extracted by a straightforward generalization of the usual [[tensor product of abelian groups]] (or of [[commutative monoids]]), where ``[[bilinear map|bilinearity]] conditions'' = ``linearity in separate variables'' is replaced by ``$T$-homomorphicity in separate variables'', where $T$ is the [[monad]] of the algebraic theory.

In slightly more detail, if $A$ and $B$ are $T$-algebras, the tensor product $A \otimes B$ ought to be $T(A \times B)$ modulo equivalences which we may write suggestively as

\begin{displaymath}
\omega(a_1, \ldots, a_m) \otimes \chi(b_1, \ldots, b_n) \sim (\omega(\chi, \ldots, \chi); a_1 \otimes b_1, \ldots, a_m \otimes b_n)
\end{displaymath}
where the left side is represented by a composite

\begin{displaymath}
T A \times T B \stackrel{\xi_A \times \xi_B}{\to} A \times B \stackrel{u(A \times B)}{\to} T(A \times B)
\end{displaymath}
(the $\xi$`s are $T$-algebra structures), and the right side by the monoidal structure map on $T$,

\begin{displaymath}
\alpha_{A, B} \colon T A \times T B \to T(A \times B).
\end{displaymath}
In more detail still, $A \otimes B$ is the following [[coequalizer]] in $Alg_T$:

\begin{displaymath}
\itexarray{
T(T A \times T B) & \stackrel{T\alpha}{\to} & T T(A \times B) & \\
 & ^\mathllap{T(\xi_A \times \xi_B)} \searrow & \downarrow^\mathrlap{m} & \\
 & & T(A \times B) & \to & A \otimes B
}
\end{displaymath}
(\hyperlink{Seal12}{Seal 12, section 2.2 and theorem 2.5.5})

This construction carries over to the wider context of monoidal monads.

\hypertarget{references}{}\subsection*{{References}}\label{references}

The notion of commutative algebraic theory was formulated in terms of [[monads]] by [[Anders Kock]].

\begin{itemize}%
\item [[Anders Kock]], \emph{Monads on symmetric monoidal closed categories}, Arch. Math. 21 (1970), 1--10.

\end{itemize}
The [[closed category]]-structure on the EM-category of monoidal monads was studied in

\begin{itemize}%
\item [[Anders Kock]], \emph{Strong functors and monoidal monads}, Arhus Universitet, Various Publications Series No. 11 (1970). \href{http://home.imf.au.dk/kock/SFMM.pdf}{PDF}.


\item [[Anders Kock]], \emph{Closed categories generated by commutative monads} ([[KockMonoidalMonads.pdf:file]])



\end{itemize}
and the [[monoidal category]]-structure in

\begin{itemize}%
\item [[William Keigher]], \emph{Symmetric monoidal closed categories generated by commutative adjoint monads}, Cahiers de Topologie et G\'e{}om\'e{}trie Diff\'e{}rentielle Cat\'e{}goriques, 19 no. 3 (1978), p. 269-293 (\href{http://www.numdam.org/numdam-bin/fitem?id=CTGDC_1978__19_3_269_0}{NUMDAM}, \href{http://archive.numdam.org/article/CTGDC_1978__19_3_269_0.pdf}{pdf})


\item Gavin J. Seal, \emph{Tensors, monads and actions} (\href{http://arxiv.org/abs/1205.0101}{arXiv:1205.0101})



\end{itemize}
There is \href{http://mathoverflow.net/a/75929/381}{related MO discussion}.

The Kronecker product of theories was introduced in an article of Freyd:

\begin{itemize}%
\item [[Peter Freyd]], \emph{Algebra valued functors in general and tensor products in particular}, Colloq. Math. 14 (1966), 89-106.

\end{itemize}
Recently [[Nikolai Durov]] rediscovered that notion for the purposes of geometry (under the name \textbf{commutative algebraic monad}), constructed their spectra (generalizing the [[spectrum (geometry)|spectrum]] of Grothendieck) and theory of [[generalized scheme after Durov|generalized schemes on this basis]]. There is a generalized version of the [[Eckmann–Hilton argument]] concerning commutative finitary monads. Much detail including many examples and further constructions are in his thesis

\begin{itemize}%
\item [[Nikolai Durov]], \emph{New approach to Arakelov geometry}, \href{http://arXiv.org/abs/0704.2030}{arXiv:0704.2030}

\end{itemize}
[[!redirects commutative algebraic theories]]

[[!redirects commutative theory]] [[!redirects commutative theories]]

[[!redirects commutative monad]] [[!redirects commutative algebraic monad]] [[!redirects commutative monads]] [[!redirects commutative algebraic monads]]



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